The walking isomorphism or free-standing isomorphism or interval groupoid is the category (in fact a groupoid) which ‘represents’ isomorphisms in a category.
The word “walking” is because it’s a “walking structure”.
The walking isomorphism can be categorified in a number of ways: to the walking equivalence, to the walking adjoint equivalence, or to the walking 2-isomorphism?. Related, though not quite a categorification in one of the usual senses, is the walking 2-isomorphism with trivial boundary.
The free-standing isomorphism is the unique (up to isomorphism) category with exactly two objects and , exactly one arrow , exactly one arrow , and no other arrows which are not identity arrows.
The arrow is an isomorphism, whose inverse is the arrow .
The free-standing isomorphism is a groupoid.
The free-standing isomorphism can also be described as the free groupoid on the interval category, that is to say, the walking arrow. Because it is an interval object for Cat and Grpd, it is also known as the interval groupoid.
Let be a category. Let denote the free-standing isomorphism. Evaluation at the arrow establishes a natural bijective correspondence between functors and isomorphisms in . Thus, for any isomorphism of there is a unique functor such that the arrow of maps under to .
Immediate from the definitions.
is the full subcategory classifier of the 1-category . That is, for any small category , there is a bijective correspondence between full subcategories of and functors , with the reverse direction given by taking the pullback of the inclusion .
Functors into are uniquely determined by functions on the sets of objects. is a full subcategory of , and any pullback of a full subcategory can be given as a full subcategory.
In fact, this generalizes. If is a simplicial set, say that a full subspace of is a subsimplicial set with the property there is some subset such that contains exactly the simplices whose vertices are all contained in .
The nerve is the full subspace classifier for , and thus represents the subpresheaf of full subspaces.
This can be determined from the explicit description of given by listing the vertices of a path. However, it’s more informative to observe that , where is the indiscrete space functor, which is the direct image part of the geometric embedding whose inverse image is .
This also implies is the full subcategory classifier of , the 1-category of quasi-categories, since those are given by full subspaces of simplicial sets.
Last revised on September 8, 2020 at 11:09:28. See the history of this page for a list of all contributions to it.